How to create a mesh with two shapekeys using extrude and bevel modifier?
Animate the modifier props.
To elaborate on @Gorgeous' comment.
Make your mesh, for example sake added a default circle, then grid filled. Note to self,  rotate after grid fill to match axes before adding shape keys in future,
Create your shape key from this. Super fast hacked together example, showing my lack of modelling with bevel modifier, however shows proof of concept.

Now add your modifiers to this object. Solidify and bevel.

any of the modifier properties can also be keyframed or driven.

Ugly example, but you get the concept,  with solidify thickness being driven directly by shapekey value, and some bevel modifier properties too.